Grunwald, Dirk C

Phillips, Caleb

Sicker, Douglas

Grunwald, Dirk C.

CU Scholar Institutional Repository

The Stability of The Longley-Rice Irregular Terrain Modelfor Typical ProblemsCaleb Phillips, Douglas Sicker, and Dirk GrunwaldCU-CS-1086-11 September 2011University of Colorado at BoulderTechnical Report CU-CS-1086-11Department of Computer ScienceCampus Box 430University of ColoradoBoulder, Colorado 80309The Stability of The Longley-Rice Irregular Terrain Modelfor Typical ProblemsCaleb Phillips, Douglas Sicker, and Dirk GrunwaldSeptember 2011AbstractIn this paper, we analyze the numerical stability of the popular Longley-Rice Irregular TerrainModel (ITM). This model is widely used to plan wireless networks and in simulation-validatedresearch and hence its stability is of fundamental importance to the correctness of a large amountof work. We take a systematic approach by first porting the reference ITM implementation to amultiprecision framework and then generating loss predictions along many random paths using realterrain data. We find that the ITM is not unstable for common numerical precisions and practicalprediction scenarios.1 IntroductionThe Irregular Terrain Model (ITM) is a well known and widely used model for predicting propagationloss in long (greater than one kilometer) outdoor radio links. This model, in its most widely usedincarnation, was developed by Hufford et al. in [1] at the National Telecommunications and InformationAdministration (NTIA) Institute for Telecommunications Sciences (ITS) based on work done in 1978 byLongley [2]. The model predicts the median attenuation of the radio signal as a function of distance andadditional losses due to refractions at intermediate (terrain) obstacles. Compared to the vast majorityof other models, even those that are similar in approach (e.g., The International TelecommunicationsUnion (ITU) Terrain Model [3]), the ITM is very complicated, requiring the interaction of dozens offunctions that implement numerical approximations to theory. Despite its age, the ITM maintains asubstantial popularity, possibly due to the fact that it was extensively validated in the 1970s for planninganalog television transmissions, and that a reference implementation is available. Today, it is used inseveral popular coverage mapping and network planning tools (e.g., [4, 5]) as well as network simulationapplications (e.g., [6, 7]). As a result, it is also used widely in current wireless networking research (e.g.,[8, 9]), emerging government standards (e.g., [10]), and persists at the foundation of other modern pathloss models such as ITU-R 452 [11]. Due to the ITM’s complexity and its simultaneous popularity, thequestion of numerical stability is an obviously important one, but to our knowledge has not previouslybeen investigated.We take a systematic empirical approach to the analysis that involves porting the defacto C++implementation of the ITM [12] to a multiprecision framework. A comparison is made between thepredicted path loss values for many randomly generated links over real terrain data. Model parame-ters are also varied in order to produce a fully factorial experimental design over a range of realisticparameters. In the end, the results show that while the model performs disastrously for half-precision(16 bit) arithmetic, it is well behaved for single-precision (64 bit) and higher precisions. Within thevalues tested, there are only very few isolated cases that result in significantly different (greater than 3dB) output and these tend to result from a single change in branching decision in the approximationalgorithms and not because of massive information loss. While this sort if empirical analysis cannot beused to extrapolate to any parameters and any terrain model, we are able to say that over realistic linksthe model appears to be well behaved. This result provides confidence in the stability of the output ofthe ITM model as well as other similar models that compute diffraction over terrain (e.g, [3, 11]).1Frequency (MHz) 0.148, 80, 900, 1900, 2400, 5280, 60000Climate Code 1 (Equatorial), 2 (Continental Subtropical),3 (Maritime Subtropical), 4 (Desert),5 (Continental Temperate), 6 (MaritimeTemperate on land), 7 (MaritimeTemperate at Sea)Permittivity/Conductivity 5/0.001 (Poor Ground), 13/0.002,15/0.005 (Average Ground),25/0.02 (Good Ground), 80/5.0(Sea Water)Tx/Rx Height (Meters) Uniform random: U(0, 35)Tx/Rx Location Uniform random within bounding box.Precision (Bits) 11, 24, 53, 64, 128, 256, 512, 1024,32-bit Intel nativeTable 1: Experimental parameters2 ImplementationThe implementation involves a line by line port of the reference ITM implementation to have multi-precision support. By and large, this involves using multiprecision data structures in place of nativemachine number formats. To this end, we make use of the MPL, MPFR, and MPC libraries [13, 14, 15].The MPFR library wraps the MPL library and provides additional necessary features such as a squareroot function, computation of logs and powers, and trigonometric functions. The MPC library providessupport for complex arithmetic. In porting, a line like:fhtv=0.05751*x-4.343*log(x);Must be replaced with:mpfr_log(tmp,x,R);mpfr_mul_d(tmp,tmp,4.343,R);mpfr_mul_d(fhtv,x,0.05751,R);mpfr_sub(fhtv,fhtv,tmp,R);Although it may be possible to automate this process, we proceeded manually in order to avoid introduc-ing bugs. In porting, the refernce ITM source is modified to take an additional command line argumentthat specifies the precision in bits, which is passed to the multiprecision framework. Otherwise, thefunctionality and usage is identical to the machine-precision ITM implementation1.2.1 ExperimentOur experimental design involves generating random link geometries within a latitude and longitudebounding box. For each random link, a path loss prediction is made both with the machine precision(64-bit double precision arithmetic) reference implementation and multiprecision implementation (at avariety of precisions). After the fact, we can quantify the differences in predictions and investigate anyoutliers or general trends.For the bounding box we use 39.95324 to 40.07186 latitude and -105.31843 to -105.18602 longitudein the WGS84 datum. This box contains a portion of the mountainous region to the west of Boulder,Colorado, as well as the plains to the east, providing a realistic mix of topographies. 500 links aregenerated uniformly at random within the box. For each random link, we try a range of reasonable1Our multiprecision implementation is available at http://systems.cs.colorado.edu/research/wireless/.2Figure 1: Terrain (Digital Elevation Model) data for area within geographic bounding box. The plottedgreyscale color varies from white (1562.15 m) to black (2550.28 m). Rolling hills and mesas (light grey)are at ≈ 1700 m, or 140m above the Colorado plateau.3parameter values, which are summarized in table 1. Antenna heights are selected uniformly at randombetween 0 and 35 meters. For each link, the corresponding elevation profile is extracted from a publiclyavailable United States Geological Survey (USGS) Digital Elevation Model (DEM) with 0.3 arcsecondraster precision. Extraction and coordinate transformations are performed with the Geospatial DataAbstraction Library (GDAL) [16]. By trying each unique combination of parameters with a random linkand random transmitter heights, we must make 122,500 predictions at each precision. We evaluate 8precisions ranging from half-precision (11 bits mantissa, 16 bits total) to 1024 bits along with the native32-bit (64 bit double precision arithmetic) machine precision. This results in 1,102,500 predictions totaltotal. Although this takes some time to compute, it can be parallelized trivially.3 ResultsFigure 2a shows the overall results of this experiment: the error (ǫ) between the multiprecision predictionand the machine precision prediction is plotted. Half-precision arithmetic (11 bits of mantissa, 16 bitstotal) produces results that vary wildly. Above this, however, beginning at single precision (23 bitsof mantissa, 1 bit of sign, 32 bits total), the two programs make very similar predictions. Figure 2bprovides a more detailed picture of these remaining cases. Much of the small error is negligible as it ispresumably a function of differences in rounding2. In the results, there is one clear outlier that producesa 6 dB difference. We investigated this case and found that it was the result of a difference in a branchingdecision that chooses whether or not to make a particular correction. It is not clear that one directiondown the branch offers a better prediction than another, so this case can be safely ignored.Finally, we investigated the performance, in terms of running time, for the various precisions (on anotherwise unloaded machine). The multiprecision version is not substantially slower than the machineprecision implementation—nearly all precisions take the same amount of time to run. At each precision,there are some number of outliers, which require slightly more time, but it is not clearly a function ofprecision. Hence, if it were the case that the multiple precision implementation was also safer, then itsuse would be clearly preferable.4 ConclusionsAlthough it is not possible to extrapolate universally from these results, they demonstrate that theITM is not substantially unstable for typical problems and reasonably precise numeric types (i.e., singleand double precision IEEE formats). An analytical investigation of stability would go a long way todetermine the stability universally, but is a substantial undertaking that involves the careful dissection ofdozens of complex algorithms that combine to create the ITM implementation. A motivated investigatormay choose to focus his effort on the knife-edge diffraction approximation algorithm, which is almostcertainly the most numerically complex component of the model. For most practical purposes, however,the results presented here are sufficient to justify continued use of this model with the confidence thatunder typical situations it is not significantly affected by rounding and cancellation errors.2IEEE 754-2008 requires subnormal arithmetic rounding, which is not done natively by the MPFR library. The majorityof rounding (excluding this special case) are identical.4Loss Value Prediction By ITM as a function of PrecisionPrecision (bits), 0 is Reference/IEEE−64/Machine PrecisionPath Loss Error due to Precision (dBm)0501001502000 11 24 53 64 128 256 512 1024(a) All PrecisionsLoss Value Prediction By ITM as a function of PrecisionPrecision (bits), 0 is Reference/IEEE−64/Machine PrecisionPath Loss Error due to Precision (dBm)012345624 53 64 128 256 512 1024(b) Single-Precision and GreaterFigure 2: Box and whiskers plot of error as a function of precision. The zero-bit case is the machine-precision reference implementation.5References[1] G. Hufford, A. Longley, and W. Kissick, “A guide to the use of the ITS irregular terrain model inthe area prediction mode,” NTIA, Tech. Rep. 82-100, 1982.[2] A. Longley, “Radio propagation in urban areas,” in Vehicular Technology Conference, 1978. 28thIEEE, vol. 28, march 1978, pp. 503 – 511.[3] J. S. Seybold, Introduction to RF Propagation. Wiley Interscience, 2005.[4] R. Coudé, “Radio Mobile,” http://www.cplus.org/rmw/english1.html, July 2010.[5] J. A. Magliacane, “SPLAT! A Terrestrial RF Path Analysis Application for Linux/Unix,” March2008. [Online]. Available: http://www.qsl.net/kd2bd/splat.html[6] Scalable Network Technologies, “Qualnet,” http://www.scalable-networks.com/products/qualnet/,August 2011.[7] OPNET Technologies, Inc., “Opnet,” http://www.opnet.com/, August 2011.[8] K. Harrison, S. M. Mishra, and A. Sahai, “How much white-space capacity is there?” in DySPAN’10,2010.[9] M. Zennaro, A. Bagula, D. Gascon, and A. Noveleta, “Planning and deploying long distancewireless sensor networks: The integration of simulation and experimentation,” in Ad-Hoc,Mobile and Wireless Networks, ser. Lecture Notes in Computer Science, I. Nikolaidis andK. Wu, Eds. Springer Berlin / Heidelberg, 2010, vol. 6288, pp. 191–204. [Online]. Available:http://dx.doi.org/10.1007/978-3-642-14785-2 15[10] C. Martin, the Commissioners Copps, Adelstein, McDowell, and Tate, “In the matter of unliscensedoperation in the TV broadcast bands. additional spectrum for unlicensed devices below 900 MHzand in the 3 GHz band.” FCC, Tech. Rep. FCC 08-260, 2008.[11] ITU-R, “Prediction procedure for the evaluation of microwave interference between stations on thesurface of the earth at frequencies above about 0.7 ghz,” ITU, Tech. Rep. P.452, 2007.[12] G. Hufford, “The ITS irregular terrain model, version 1.2.2, the algorithm,” http://flattop.its.bldrdoc.gov/itm.html.[13] “The GNU MPFR Library,” http://www.mpfr.org/, June 2010.[14] “The GNU Multiple Precision Arithmetic Library,” http://gmplib.org/, June 2010.[15] “MPC,” http://www.multiprecision.org/, June 2010.[16] T. O. S. G. Foundation, “Geospatial Data Abstraction Library (GDAL),” http://www.gdal.org/.6

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